Basic Math Examples

$3 y^{2} - y - 9 = 0$

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = 3$ , $b = - 1$ , and $c = - 9$ into the quadratic formula and solve for $y$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (3 \cdot - 9)}}{2 \cdot 3}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 1$ to the power of $2$ .

$y = \frac{1 \pm \sqrt{1 - 4 \cdot 3 \cdot - 9}}{2 \cdot 3}$

Step 3.1.2

Multiply $- 4 \cdot 3 \cdot - 9$ .

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Step 3.1.2.1

Multiply $- 4$ by $3$ .

$y = \frac{1 \pm \sqrt{1 - 12 \cdot - 9}}{2 \cdot 3}$

Step 3.1.2.2

Multiply $- 12$ by $- 9$ .

$y = \frac{1 \pm \sqrt{1 + 108}}{2 \cdot 3}$

Step 3.1.3

Add $1$ and $108$ .

$y = \frac{1 \pm \sqrt{109}}{2 \cdot 3}$

Step 3.2

Multiply $2$ by $3$ .

$y = \frac{1 \pm \sqrt{109}}{6}$

Step 4

The final answer is the combination of both solutions.

$y = \frac{1 + \sqrt{109}}{6}, \frac{1 - \sqrt{109}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{1 + \sqrt{109}}{6}, \frac{1 - \sqrt{109}}{6}$

Decimal Form:

$y = 1.90671775 \dots, - 1.57338441 \dots$